The derivative of the fractional discrete Laplacian is an exotic Riesz potential

TL;DR

This paper investigates the derivatives of the fractional discrete Laplacian on integer lattices, revealing they are exotic Riesz potentials with specific properties, extending continuous analogs to the discrete setting.

Contribution

It establishes that derivatives of the fractional discrete Laplacian are exotic Riesz potentials, extending the logarithmic Laplacian concept to discrete spaces.

Findings

Right hand derivative at 0 is an exotic discrete Riesz potential for N=1.

For N≥2, the derivative includes an additional corrector term.

Results extend the logarithmic Laplacian to the discrete lattice setting.

Abstract

Let $\Delta_{N}$ be the multidimensional discrete Laplacian on $\mathbb{Z}^N$ ($N\ge1$). In this note, we prove that, when $N=1$, the right hand derivative of $(-\Delta_1)^s$ at $0$ is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math. 2000), and when $N\ge 2$, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Comm. PDEs. 2019) to the discrete setting.